Subgroups — C₆₇×#22145 #30701396

Order 17152.

1064 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	—	2	1	no	—
15	C₂×C₄×D₄×C₆₇	—	2	1	no	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	—	2	1	no	—
19	#459×C₆₇	C₄	1	8	yes	complemented
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	#390×C₆₇	—	2	1	no	—
29	#390×C₆₇	—	2	1	no	—
30	#390×C₆₇	C₂²	1	1	yes	characteristic
31	#390×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	2	yes	—
33	#389×C₆₇	C₂²	1	1	yes	characteristic
34	#389×C₆₇	C₂²	1	1	yes	characteristic
35	#388×C₆₇	—	2	1	no	—
36	#388×C₆₇	—	2	1	no	—
37	#384×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂³×C₄×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	—	4	2	no	—
51	C₂×D₈×C₆₇	—	4	2	no	—
52	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	#127×C₆₇	—	2	2	no	—
56	#127×C₆₇	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	4	2	no	—
59	C₄×D₄×C₆₇	—	4	2	no	—
60	C₄×D₄×C₆₇	—	4	4	no	—
61	C₄×D₄×C₆₇	—	4	4	no	—
62	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
63	#117×C₆₇	—	4	4	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	#107×C₆₇	—	4	2	no	—
89	C₄⋊C₈×C₆₇	—	2	2	no	—
90	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
91	#102×C₆₇	—	4	2	no	—
92	#102×C₆₇	—	4	2	no	—
93	#102×C₆₇	—	4	2	no	—
94	#102×C₆₇	C₂×C₄	1	8	yes	complemented
95	C₄×C₈×C₆₇	—	4	2	no	—
96	#95×C₆₇	—	2	1	no	—
97	#95×C₆₇	—	2	1	no	—
98	#95×C₆₇	—	2	1	no	—
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂⁴×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	—	4	2	no	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	4	4	no	—
125	C₂²×C₄×C₆₇	—	4	4	no	—
126	C₂²×C₄×C₆₇	—	4	2	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	—	4	4	no	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
136	D₈×C₆₇	—	4	8	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
143	C₄⋊C₄×C₆₇	—	2	1	no	—
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	4	2	no	—
151	C₄⋊C₄×C₆₇	—	4	2	no	—
152	C₄⋊C₄×C₆₇	—	4	2	no	—
153	C₄⋊C₄×C₆₇	—	4	2	no	—
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂²⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	2	no	—
157	C₂²⋊C₄×C₆₇	—	4	4	no	—
158	C₂²⋊C₄×C₆₇	—	4	2	no	—
159	C₂²⋊C₄×C₆₇	—	4	2	no	—
160	C₂²⋊C₄×C₆₇	—	4	4	no	—
161	C₂²⋊C₄×C₆₇	—	4	2	no	—
162	C₂²⋊C₄×C₆₇	—	4	4	no	—
163	C₄²×C₆₇	—	2	2	no	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	1	no	—
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	4	1	no	—
170	C₄²×C₆₇	—	4	2	no	—
171	C₄²×C₆₇	D₈	1	4	yes	complemented
172	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
173	C₄²×C₆₇	#41	1	2	yes	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	4	2	no	—
177	C₂³×C₆₇	—	4	1	no	—
178	C₂³×C₆₇	—	4	2	no	—
179	C₂³×C₆₇	—	4	1	no	—
180	C₂³×C₆₇	—	4	1	no	—
181	C₂³×C₆₇	—	4	2	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	—	4	2	no	—
185	C₂³×C₆₇	—	4	1	no	—
186	C₂³×C₆₇	—	4	1	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	D₄×C₆₇	—	2	2	no	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	2	2	no	—
192	D₄×C₆₇	—	2	4	no	—
193	D₄×C₆₇	—	4	8	no	—
194	D₄×C₆₇	—	4	1	no	—
195	D₄×C₆₇	—	4	2	no	—
196	D₄×C₆₇	—	4	1	no	—
197	D₄×C₆₇	—	4	2	no	—
198	D₄×C₆₇	—	4	8	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	1	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	8	8	no	—
224	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
227	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
228	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
229	C₂×C₄×C₆₇	#135	1	2	yes	—
230	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
231	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
237	C₈×C₆₇	—	2	4	no	—
238	C₈×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	4	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	4	2	no	—
245	C₂²×C₆₇	—	8	2	no	—
246	C₂²×C₆₇	—	8	1	no	—
247	C₂²×C₆₇	—	8	2	no	—
248	C₂²×C₆₇	—	8	1	no	—
249	C₂²×C₆₇	—	8	1	no	—
250	C₂²×C₆₇	—	8	2	no	—
251	C₂²×C₆₇	—	8	1	no	—
252	C₂²×C₆₇	—	8	2	no	—
253	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂²×C₆₇	#535	1	1	yes	characteristic
255	C₂²×C₆₇	#537	1	1	yes	characteristic
256	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
257	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
258	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂²×C₆₇	#575	1	1	yes	characteristic
260	C₄×C₆₇	—	2	2	no	—
261	C₄×C₆₇	—	2	2	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	4	2	no	—
265	C₄×C₆₇	—	4	1	no	—
266	C₄×C₆₇	—	8	1	no	—
267	C₄×C₆₇	—	8	2	no	—
268	C₄×C₆₇	—	8	1	no	—
269	C₄×C₆₇	C₄×D₈	1	4	yes	—
270	C₄×C₆₇	#518	1	1	yes	characteristic
271	C₄×C₆₇	#520	1	1	yes	characteristic
272	C₄×C₆₇	#540	1	1	yes	characteristic
273	C₄×C₆₇	#542	1	1	yes	characteristic
274	#22145	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂×C₆₇	—	8	2	no	—
276	C₂×C₆₇	—	8	4	no	—
277	C₂×C₆₇	#2324	1	1	yes	characteristic
278	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
279	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₆₇	#3205	1	1	yes	characteristic
281	C₂×C₆₇	#3212	1	1	yes	characteristic
282	C₂×C₆₇	#3245	1	1	yes	characteristic
283	C₂×C₆₇	#3247	1	1	yes	characteristic
284	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
285	C₄×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#2293	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1970	C₂×C₆₇	1	2	yes	complemented
289	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
290	#1934	C₂×C₆₇	1	2	yes	complemented
291	#1879	C₂×C₆₇	1	2	yes	complemented
292	#1862	C₂×C₆₇	1	2	yes	complemented
293	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
295	C₆₇	#22145	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
297	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×D₄	—	2	1	no	—
299	C₂×C₄×D₄	—	2	1	no	—
300	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
302	C₂×#117	—	2	1	no	—
303	#459	C₄×C₆₇	1	8	yes	complemented
304	C₂×#107	C₂²×C₆₇	1	2	yes	—
305	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×#102	C₂²×C₆₇	1	2	yes	—
308	C₂×#102	C₂²×C₆₇	1	2	yes	—
309	C₂×#102	C₂²×C₆₇	1	2	yes	—
310	C₂×#102	C₂²×C₆₇	1	2	yes	—
311	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
312	#390	—	2	1	no	—
313	#390	—	2	1	no	—
314	#390	C₂²×C₆₇	1	2	yes	—
315	#390	C₂²×C₆₇	1	1	yes	characteristic
316	#390	C₂²×C₆₇	1	1	yes	characteristic
317	#389	C₂²×C₆₇	1	1	yes	characteristic
318	#389	C₂²×C₆₇	1	1	yes	characteristic
319	#388	—	2	1	no	—
320	#388	—	2	1	no	—
321	#384	C₂²×C₆₇	1	1	yes	characteristic
322	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
323	C₄×C₂²⋊C₄	—	2	1	no	—
324	C₄³	C₂²×C₆₇	1	1	yes	characteristic
325	#340	C₂²×C₆₇	1	2	yes	—
326	#340	C₂²×C₆₇	1	2	yes	—
327	#336	C₂²×C₆₇	1	2	yes	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
331	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂×D₈	—	4	2	no	—
335	C₂×D₈	—	4	2	no	—
336	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
337	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
338	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
339	#127	—	2	2	no	—
340	#127	C₂×C₄×C₆₇	1	4	yes	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	4	2	no	—
343	C₄×D₄	—	4	2	no	—
344	C₄×D₄	—	4	4	no	—
345	C₄×D₄	—	4	4	no	—
346	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
347	#117	—	4	4	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
372	#107	—	4	2	no	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
375	#102	—	4	2	no	—
376	#102	—	4	2	no	—
377	#102	—	4	2	no	—
378	#102	C₂×C₄×C₆₇	1	8	yes	complemented
379	C₄×C₈	—	4	2	no	—
380	#95	—	2	1	no	—
381	#95	—	2	1	no	—
382	#95	—	2	1	no	—
383	C₂⁴	—	2	1	no	—
384	C₂⁴	—	2	2	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	4	2	no	—
398	C₂×D₄	—	4	2	no	—
399	C₂×D₄	—	4	2	no	—
400	C₂×D₄	—	4	2	no	—
401	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
402	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	—	4	4	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
420	D₈	—	4	8	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	—	4	2	no	—
436	C₄⋊C₄	—	4	2	no	—
437	C₄⋊C₄	—	4	2	no	—
438	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	4	4	no	—
441	C₂²⋊C₄	—	4	2	no	—
442	C₂²⋊C₄	—	4	2	no	—
443	C₂²⋊C₄	—	4	2	no	—
444	C₂²⋊C₄	—	4	4	no	—
445	C₂²⋊C₄	—	4	4	no	—
446	C₂²⋊C₄	—	4	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	—	4	2	no	—
455	C₄²	D₈×C₆₇	1	4	yes	complemented
456	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
457	C₄²	#41×C₆₇	1	2	yes	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	2	no	—
472	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
473	D₄	—	2	2	no	—
474	D₄	—	2	4	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	4	no	—
477	D₄	—	4	8	no	—
478	D₄	—	4	1	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	1	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	8	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	8	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	8	8	no	—
508	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
509	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
510	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
511	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
512	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
513	C₂×C₄	#135×C₆₇	1	2	yes	—
514	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
517	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
518	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
519	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
520	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
521	C₈	—	2	4	no	—
522	C₈	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	8	2	no	—
530	C₂²	—	8	1	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	1	no	—
533	C₂²	—	8	1	no	—
534	C₂²	—	8	2	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	2	no	—
537	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
538	C₂²	#535×C₆₇	1	1	yes	characteristic
539	C₂²	#537×C₆₇	1	1	yes	characteristic
540	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
541	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
542	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
543	C₂²	#575×C₆₇	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	4	2	no	—
549	C₄	—	4	1	no	—
550	C₄	—	8	2	no	—
551	C₄	—	8	1	no	—
552	C₄	—	8	1	no	—
553	C₄	C₄×D₈×C₆₇	1	4	yes	—
554	C₄	#518×C₆₇	1	1	yes	characteristic
555	C₄	#520×C₆₇	1	1	yes	characteristic
556	C₄	#540×C₆₇	1	1	yes	characteristic
557	C₄	#542×C₆₇	1	1	yes	characteristic
558	C₂	—	8	2	no	—
559	C₂	—	8	4	no	—
560	C₂	C₆₇×#2324	1	1	yes	characteristic
561	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
562	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
563	C₂	C₆₇×#3205	1	1	yes	characteristic
564	C₂	C₆₇×#3212	1	1	yes	characteristic
565	C₂	C₆₇×#3245	1	1	yes	characteristic
566	C₂	C₆₇×#3247	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#127×C₆₇	C₂	semidirect
3. C₆₇×#2293	C₂	semidirect
4. C₆₇×#1982	C₂	semidirect
5. C₆₇×#1970	C₂	semidirect
6. C₆₇×#1947	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1879	C₂	semidirect
9. C₆₇×#1862	C₂	semidirect
10. C₆₇×#1802	C₂	semidirect
11. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
19. #459×C₆₇	C₄	semidirect
52. C₂×D₈×C₆₇	C₂×C₄	semidirect
62. C₄×D₄×C₆₇	C₂×C₄	semidirect
90. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
94. #102×C₆₇	C₂×C₄	semidirect
171. C₄²×C₆₇	D₈	semidirect
274. #22145	C₆₇	direct (N × H)
285. C₄×#127	C₂×C₆₇	semidirect
286. #2293	C₂×C₆₇	semidirect
287. #1982	C₂×C₆₇	semidirect
288. #1970	C₂×C₆₇	semidirect
289. #1947	C₂×C₆₇	semidirect
290. #1934	C₂×C₆₇	semidirect
291. #1879	C₂×C₆₇	semidirect
292. #1862	C₂×C₆₇	semidirect
293. #1802	C₂×C₆₇	semidirect
294. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
295. C₆₇	#22145	direct (N × H)
303. #459	C₄×C₆₇	semidirect
336. C₂×D₈	C₂×C₄×C₆₇	semidirect
346. C₄×D₄	C₂×C₄×C₆₇	semidirect
374. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
378. #102	C₂×C₄×C₆₇	semidirect
455. C₄²	D₈×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#459×C₆₇	2
C₄×#127×C₆₇	2
C₆₇×#2293	2
C₆₇×#1982	2
C₆₇×#1970	2
C₆₇×#1947	2
C₆₇×#1934	2
C₆₇×#1879	2
C₆₇×#1862	2
C₆₇×#1802	2
C₄×C₄⋊C₈×C₆₇	2
#22145	67

Subgroups — C67×#22145 #30701396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22145 #30701396